Barnes G-function - Asymptotic Expansion

Asymptotic Expansion

The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:

$\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right).$

Here the are the Bernoulli numbers and is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes the Bernoulli number would have been written as, but this convention is no longer current.) This expansion is valid for in any sector not containing the negative real axis with large.

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